Liouville theorems and gradient estimates of a nonlinear elliptic equation for the V-Laplacian

TL;DR

This paper derives gradient estimates, Liouville theorems, and Harnack inequalities for positive solutions of a nonlinear elliptic equation involving the V-Laplacian on smooth metric measure spaces with curvature bounds.

Contribution

It extends existing results by establishing new gradient estimates and Liouville theorems for a broad class of nonlinear elliptic equations on curved spaces.

Findings

Gradient estimates for solutions on curved spaces

Liouville theorems under curvature bounds

Harnack inequalities for the nonlinear equation

Abstract

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is bounded from below by $-(k-1)K$ with $K \geq 0$. Additionally, we obtain related Liouville theorems and Harnack inequalities. We partially extend conclusions of Wang, when $V=0$, $\mu=0$ the equation becomes $\Delta u^{m}+p(x)u^{\alpha}=0$. And $V=f$, $\mu=c, p=0 $, the equation becomes $\Delta_{f}u^{m}+cu=0 $.